import numpy as np
import matplotlib.pyplot as plt
%matplotlib notebook
vmax = 0.75
goal_threshold = 0.05
lookahead = 3.0
#You know what to do!
def simulate_unicycle(pose, v,w, dt=0.1):
x, y, t = pose
return x + np.cos(t)*dt, y + np.sin(t)*dt, t+w*dt
class PurePursuitTracker(object):
def __init__(self, x, y, v, lookahead = 3.0):
"""
Tracks the path defined by x, y at velocity v
x and y must be numpy arrays
v and lookahead are floats
"""
self.length = len(x)
self.ref_idx = 0 #index on the path that tracker is to track
self.lookahead = lookahead
self.x, self.y = x, y
self.v, self.w = v, 0
def update(self, xc, yc, theta):
"""
Input: xc, yc, theta - current pose of the robot
Update v, w based on current pose
Returns True if trajectory is over.
"""
#Calculate ref_x, ref_y using current ref_idx
#Check if we reached the end of path, then return TRUE
#Two conditions must satisfy
#1. ref_idx exceeds length of traj
#2. ref_x, ref_y must be within goal_threshold
# Write your code to check end condition
ref_x, ref_y = self.x[self.ref_idx], self.y[self.ref_idx]
goal_x, goal_y = self.x[-1], self.y[-1]
if (self.ref_idx > self.length) or \
(np.linalg.norm([ref_x-goal_x, ref_y-goal_y])) < goal_threshold:
return True
#End of path has not been reached
#update ref_idx using np.hypot([ref_x-xc, ref_y-yc]) < lookahead
if np.hypot(ref_x-xc, ref_y-yc) < lookahead:
self.ref_idx = self.ref_idx + 1
#Find the anchor point
# this is the line we drew between (0, 0) and (x, y)
anchor = np.asarray([ref_x - xc, ref_y - yc])
#Remember right now this is drawn from current robot pose
#we have to rotate the anchor to (0, 0, pi/2)
#code is given below for this
theta = np.pi/2 - theta
rot = np.asarray([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])
anchor = np.dot(rot, anchor)
L = np.sqrt(anchor[0] ** 2 + anchor[1] **2) # dist to reference path
X = anchor[0] #cross-track error
#from the derivation in notes, plug in the formula for omega
self.w = -1*2*self.v*X/L**2
return False
x = np.arange(0, 50, 0.5)
y = [np.sin(idx / 5.0) * idx / 2.0 for idx in x]
#write code here
target_speed = 10.0 / 3.6 # [m/s]
#write code to instantiate the tracker class
tracker = PurePursuitTracker(x,y,target_speed)
pose = -1, 0, np.pi/2 #arbitrary initial pose
x0,y0,t0 = pose # record it for plotting
traj =[]
while True:
#write the usual code to obtain successive poses
pose = simulate_unicycle(pose,tracker.v,tracker.w)
if tracker.update(*pose):
print("ARRIVED!!")
break
traj.append([*pose, tracker.w, tracker.ref_idx])
print(traj)
ARRIVED!! [[-1.0, 0.1, 1.5707963267948966, -5.500550055005501, 1], [-1.0, 0.2, 1.0207413212943464, -3.337864590197384, 2], [-0.947726587832202, 0.28524957700971604, 0.686954862274608, -2.001179836401396, 3], [-0.8704085098760229, 0.3486681454353926, 0.48683687863446834, -1.204012471451654, 4], [-0.782026800984548, 0.3954514066879237, 0.3664356314893029, -0.7195251865461889, 5], [-0.6886657658784027, 0.4312804050538995, 0.29448311283468404, -0.41668825588076797, 5], [-0.5929705364588261, 0.46030493030987246, 0.25281428724660726, -0.3703893187219352, 5], [-0.4961493044759516, 0.48531790768799343, 0.21577535537441372, -0.32763247585479316, 6], [-0.39846823642785745, 0.5067283944873072, 0.1830121077889344, -0.13522546170430105, 6], [-0.3001382390149465, 0.5249276144919376, 0.1694895616185043, -0.12088902527402193, 6], [-0.20157113944749616, 0.5417955391674965, 0.1574006590911021, -0.10766910619117476, 6], [-0.10280733243684964, 0.5574706922692614, 0.14663374847198463, -0.0955145228101153, 6], [-0.003880480323900898, 0.5720815762610933, 0.1370822961909731, -0.0843737983916716, 6], [0.09518141230239, 0.5857469130405956, 0.12864491635180594, -0.07419550460680277, 7], [0.19435507714133748, 0.5985759505108079, 0.12122536589112566, 0.06205338903729048, 7], [0.2936211970686692, 0.6106688176056123, 0.12743070479485472, 0.05532260833404519, 7], [0.39281036596301067, 0.6233774278384349, 0.13296296562825924, 0.04913217853726232, 7], [0.49192770998583707, 0.6366345811475235, 0.13787618348198546, 0.04345517124878324, 7], [0.5909787226588453, 0.6503785575900073, 0.1422217006068638, 0.038265037543651986, 8], [0.6899690756204551, 0.6645528307825175, 0.14604820436122898, 0.13480816626645153, 8], [0.7889044660908653, 0.6791057862372515, 0.15952902098787414, 0.1204421891690358, 8], [0.8876346870319087, 0.6949911088003202, 0.1715732399047777, 0.10720115890138485, 8], [0.9861664253213516, 0.7120643788345211, 0.1822933557949162, 0.09503314995825694, 8], [1.0845094780599491, 0.7301929192626256, 0.1917966707907419, 0.08388596188348175, 9], [1.1826758113671998, 0.7492552120158532, 0.20018526697909006, 0.15392427807242437, 9], [1.2806787867826865, 0.769140302151719, 0.21557769478633249, 0.1378769229305384, 9], [1.3783640849319796, 0.790531480835553, 0.22936538707938633, 0.12305425867896316, 9], [1.4757451725733446, 0.8132674384750458, 0.24167081294728265, 0.10940262392902858, 9], [1.572839118834347, 0.8371999602699927, 0.2526110753401855, 0.09686755552079045, 9], [1.6696654317528845, 0.8621932619046593, 0.26229783089226455, 0.08539426734078492, 10], [1.7662451017267888, 0.8881233091034537, 0.27083725762634303, 0.14248348037356445, 10], [1.8625998252928477, 0.9148771358672715, 0.28508560566369945, 0.12755873952662017, 10], [1.9585635833399326, 0.9430010961410404, 0.2978414796163615, 0.11378057342921587, 10], [2.054160798286353, 0.9723468367942916, 0.30921953695928306, 0.10109766512882776, 10], [2.1494179349672153, 1.0027783650399043, 0.3193293034721658, 0.08945821210174397, 10], [2.244362553293966, 1.0341713491412547, 0.3282751246823402, 0.0788103133303894, 11], [2.3390225402682616, 1.0664124233483303, 0.3361561560153792, 0.12327435530227795, 11], [2.43342549727515, 1.0993985069042107, 0.348483591545607, 0.11033608469826331, 11], [2.5274146578492784, 1.1335458010074146, 0.3595172000154333, 0.09839629471785417, 11], [2.6210213371081683, 1.1687280351337936, 0.3693568294872187, 0.08740918839693626, 11], [2.714277310421736, 1.2048296063092867, 0.37809774832691234, 0.07732887865106326, 11], [2.807214164323735, 1.2417449308615436, 0.3858306361920187, 0.06810964389319027, 12], [2.8998627803209907, 1.2793778148035573, 0.39264160058133774, 0.10032979100803108, 12], [2.992252933126619, 1.3176408474140624, 0.4026745796821408, 0.0897920918837374, 12], [3.0842545501778567, 1.3568288871705676, 0.41165378887051457, 0.08007021969071478, 12], [3.175900585505484, 1.3968414378106768, 0.41966081083958606, 0.07112591063282818, 12], [3.2672233050765103, 1.4375865097844955, 0.4267734019028689, 0.06292109651443019, 12], [3.3582539141056253, 1.4789800868199297, 0.4330655115543119, 0.05541804659446, 13], [3.449022269953481, 1.520945615253121, 0.43860731621375787, 0.07583689231189193, 13], [3.539556668413633, 1.563413517196335, 0.44619100544494705, 0.06786301326718565, 13], [3.629766403180059, 1.6065667760926345, 0.45297730677166564, 0.060508437524729765, 13], [3.7196812119297666, 1.6503312270489023, 0.45902815052413865, 0.05374356680390245, 13], [3.809329564433343, 1.6946389339788992, 0.4644025072045289, 0.04753912297395885, 13], [3.898738497977131, 1.7394278009277577, 0.46915641950192477, 0.041866213057758256, 14], [3.987933499672703, 1.7846412023981941, 0.4733430408077006, 0.05081988677609113, 14], [4.076938428836507, 1.8302276322271063, 0.47842502948530974, 0.04545988570491289, 14], [4.1657105399319985, 1.8762657934811113, 0.482971018055801, 0.040518135544524075, 14], [4.254272445512303, 1.9227070346357715, 0.4870228316102534, 0.03597432429151293, 14], [4.342645453389127, 1.9695067299211335, 0.4906202640394047, 0.03180844283891591, 14], [4.4308495310476825, 2.016624037615903, 0.49380110832329627, 0.028000807790888035, 15], [4.518903289926189, 2.0640216699129477, 0.49660118910238504, 0.025527417373847897, 15], [4.606823986588528, 2.1116656737162494, 0.49915393083976983, 0.02281717626107402, 15], [4.694622774078883, 2.1595339608719804, 0.5014356484658772, 0.02032005977338409, 15], [4.7823121111989995, 2.2076024552881197, 0.5034676544432156, 0.018025565258985528, 15], [4.869903591881668, 2.2558490356002805, 0.5052702109691142, 0.015923373184693838, 15], [4.957407963121717, 2.3042534260404417, 0.5068625482875836, 0.014003352254280265, 16], [5.044835147341751, 2.3527970915319996, 0.5082628835130116, -0.000325285334102487, 16], [5.13219426845958, 2.401463136753699, 0.5082303549796013, -0.0002903517950092079, 16], [5.219554972566269, 2.4501263402855624, 0.5082013198001004, -0.00025819880382738997, 16], [5.306917089580983, 2.4987870072631884, 0.5081754999197177, -0.0002286873097929937, 16], [5.394280462979177, 2.547445418545183, 0.5081526311887383, -0.00020168082817144942, 17], [5.481644949110644, 2.5961018319249707, 0.5081324631059212, -0.025851778295426446, 17], [5.569010416530917, 2.644756483320671, 0.5055472852763786, -0.023110940306898947, 17], [5.656501372799326, 2.693185117115559, 0.5032361912456887, -0.02058532024302847, 17], [5.744104018443277, 2.7414114219312955, 0.5011776592213859, -0.01826432884776407, 17], [5.831805753799582, 2.78945729184217, 0.4993512263366095, -0.01613756096517679, 17], [5.919595095383694, 2.8373429003715396, 0.4977374702400918, -0.014194800750377042, 18], [6.007461598316105, 2.8850867760251617, 0.49631799016505407, -0.05380672358480123, 18], [6.095395784183819, 2.9327058788703684, 0.49093731780657396, -0.047966927858281064, 18], [6.183584918689448, 2.979851149631582, 0.48614062502074584, -0.04259644311807287, 18], [6.27199917917064, 3.026572863465343, 0.48188098070893853, -0.037671783495459445, 18], [6.360611654814413, 3.0729175412642213, 0.4781138023593926, -0.03316982136725427, 19], [6.449398089934946, 3.118928071999993, 0.47479682022266717, -0.08473266683591857, 19], [6.538336652453995, 3.164643847144304, 0.4663235535390753, -0.07554353885284737, 19], [6.62765937958386, 3.209604390046761, 0.45876919965379054, -0.06709090456704127, 19], [6.717319202600871, 3.253888880972572, 0.4520601091970864, -0.05933851913761556, 19], [6.807274114172744, 3.2975708438874474, 0.44612625728332483, -0.05225050093171663, 20], [6.897486642844255, 3.3407182617788083, 0.4409012071901532, -0.11711048539010174, 20], [6.987923386459794, 3.3833937278479356, 0.429190158651143, -0.10435246331567151, 20], [7.0788536915379465, 3.4250071826239354, 0.41875491231957584, -0.0926188659583568, 20], [7.170213284532234, 3.4656695087738028, 0.4094930257237402, -0.08186024991946585, 20], [7.261945563494532, 3.505483940785111, 0.4013070007317936, -0.07202726673632992, 21], [7.354000687233321, 3.5445461244598135, 0.3941042740581606, -0.15145196302875286, 21], [7.446334774905601, 3.5829442527179407, 0.3789590777552853, -0.13481500996197265, 21], [7.539239798081753, 3.6199396128140213, 0.36547757675908804, -0.11952028860339332, 21], [7.63263511647742, 3.6556791497692194, 0.3535255478987487, -0.10550463409987049, 22], [7.7264509139526485, 3.690299897076102, 0.34297508448876163, -0.19119509230994017, 22], [7.820626748202551, 3.7239289357679053, 0.3238555752577676, -0.1706386201835099, 22], [7.915428301266271, 3.755751341973387, 0.3067917132394166, -0.15169342704641237, 22], [8.010759039522393, 3.785951513223613, 0.29162237053477535, -0.13428994734435232, 22], [8.106536908938583, 3.8147021606962377, 0.27819337580034015, -0.11835635710854508, 23], [8.202692222983774, 3.842164053946169, 0.26635774008948565, -0.21979144365503256, 23], [8.299165823656148, 3.868485991057778, 0.2443785957239824, -0.19596475780027708, 23], [8.396194609975582, 3.892681334109234, 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[9.584516484103789, 4.037809602570665, -0.013013302242442022, -0.1677925543966033, 26], [9.684508016921518, 4.036508309075296, -0.02979255768210235, -0.28434288485244164, 26], [9.784463640379373, 4.033529494017022, -0.05822684616734652, -0.25422892477762443, 26], [9.88429416998731, 4.027710099014206, -0.08364973864510897, -0.22638898436551086, 26], [9.983944510008923, 4.019354877079567, -0.10628863708166006, -0.20074726155580966, 26], [10.083380177875107, 4.00874601493483, -0.126363363237241, -0.17721796347585672, 27], [10.182582854695676, 3.9961432806428703, -0.14408515958482668, -0.28297885059422606, 27], [10.281546622625006, 3.9817845677050885, -0.17238304464424928, -0.25402113712558994, 27], [10.380064502593216, 3.9646315118091393, -0.19778515835680827, -0.2271680573446175, 27], [10.478114922065188, 3.944981696507964, -0.22050196409127001, -0.20235457164104098, 27], [10.575693700360521, 3.923109750403645, -0.24073742125537412, -0.17950617970729188, 27], [10.672809942740196, 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[47.9571956727143, -4.002605644366345, -1.3639633872011936, -0.0002852659818973734, 98], [47.97773181037272, -4.100474265742527, -1.3639919137993832, -0.0002650310265946857, 98], [47.998265156163946, -4.198343472905035, -1.3640184169020426, -0.00024584873446912393, 98], [48.01879590811032, -4.2962132242305415, -1.3640430017754894, -0.00022768507926760841, 98], [48.039324253935035, -4.39408348027241, -1.3640657702834162, -0.00021050643588254586, 98], [48.05985037139473, -4.491954203688714, -1.3640868209270045, -0.00019427958421780838, 98], [48.08037442860816, -4.589825359171316, -1.3641062488854263, -0.00017897171317630254, 98], [48.100896584380976, -4.687696913375978, -1.364124146056744, -0.00016455042478572966, 98], [48.12141698852653, -4.785568834853503, -1.3641406010992225, -0.00015098373845083248, 98], [48.14193578218269, -4.8834410939819, -1.3641556994730677, -0.00013824009536497262, 98], [48.16245309812455, -4.9813136628995585, -1.3641695234826041, -0.00012628836305914897, 98], [48.18296906107312, -5.079186515439437, -1.36418215231891, -0.00011509784013650461, 98], [48.20348378799983, -5.177059627064248, -1.3641936621029236, -0.00010463826116265499, 98], [48.2239973884268, -5.2749329748026526, -1.3642041259290398, -9.4879801751647e-05, 98], [48.24450996472296, -5.372806537186447, -1.364213613909215, -8.579308384031225e-05, 98], [48.26502161239577, -5.470680294188753, -1.3642221932175989, -7.734918119040245e-05, 98], [48.285532420378686, -5.568554227163208, -1.364229928135718, -6.951962510103183e-05, 98], [48.30604247131413, -5.666428318784155, -1.364236880098228, -6.227641036304643e-05, 98], [48.32655184183206, -5.764302552987843, -1.3642431077392643, -5.559200148654849e-05, 98], [48.347060602823994, -5.86217691491463, -1.364248666939413, -4.943933920074325e-05, 98], [48.367568819712446, -5.960051390852212, -1.364253610873333, -4.379184724387652e-05, 98], [48.3880765527157, -6.057925968179867, -1.3642579900580574, -3.862343949980926e-05, 99]]
xs,ys,ts,ws,ids = zip(*traj)
plt.figure()
plt.plot(x,y,label='Reference')
plt.quiver(x0,y0, np.cos(t0), np.sin(t0),scale=12)
plt.plot(xs,ys,label='Tracked')
x0,y0,t0 = pose
plt.quiver(x0,y0, np.cos(t0), np.sin(t0),scale=12)
plt.title('Pure Pursuit trajectory')
plt.legend()
plt.grid()
plt.figure()
plt.title('Curvature')
plt.plot(np.abs(ws))
plt.grid()
Make a video to plot the current pose of the robot and reference pose it is trying to track. You can use funcAnimation in matplotlib
# !apt install ffmpeg
from matplotlib import animation
from matplotlib import rc
fig, ax = plt.subplots()
ax.set_xlim(-10, 50)
ax.set_ylim(-22, 22)
line, = ax.plot([], [])
def animation_frame(i):
line.set_data(xs,ys)
line.set_data(x,y)
return line,
anim = animation.FuncAnimation(fig, animation_frame, frames=np.arange(1,len(traj),2))
rc('animation', html = 'jshtml')
rc('animation', embed_limit = 4096)
#HTML(anim.to_jshtml())
fig.show()
anim
What happens if you add a bit of Gaussian noise to the simulate_unicycle() output? Is the tracker still robust?
The noise signifies that $v$, $\omega$ commands did not get realized exactly